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Abstract

This paper presents a vaccination strategy for fighting against the propagation of
epidemic diseases. The disease propagation is described by an SEIR (susceptible plus
infected plus infectious plus removed populations) epidemic model. The model takes
into account the total population amounts as a refrain for the illness transmission
since its increase makes the contacts among susceptible and infected more difficult.
The vaccination strategy is based on a continuous-time nonlinear control law synthesised
via an exact feedback input-output linearization approach. An observer is incorporated
into the control scheme to provide online estimates for the susceptible and infected
populations in the case when their values are not available from online measurement
but they are necessary to implement the control law. The vaccination control is generated
based on the information provided by the observer. The control objective is to asymptotically
eradicate the infection from the population so that the removed-by-immunity population
asymptotically tracks the whole one without precise knowledge of the partial populations.
The model positivity, the eradication of the infection under feedback vaccination
laws and the stability properties as well as the asymptotic convergence of the estimation
errors to zero as time tends to infinity are investigated.

Keywords:

1 Introduction

A relevant area in the mathematical theory of epidemiology is the development of
models for studying the propagation of epidemic diseases in a host population [1-20]. The epidemic mathematical models analysed in such an exhaustive list of books and
papers include the most basic ones [1-9], namely (i) SI models where only susceptible and infected populations are assumed
to be present in the model, (ii) SIR models which include susceptible plus infected
plus removed-by-immunity populations and (iii) SEIR models where the infected population
is split into two ones, namely the ‘infected’ (or ‘exposed’) which incubate the disease
but do not still have any disease symptoms and the ‘infectious’ (or ‘infective’) which
do have the external disease symptoms. Those models can be divided into two main classes,
namely the so-called ‘pseudo-mass action models’, where the total population is not
taken into account as a relevant disease contagious factor and the so-called ‘true-mass
action models’, where the total population is more realistically considered as an
inverse factor of the disease transmission rates. There are many variants of the above
models as, for instance, the SVEIR epidemic models which incorporate the dynamics
of a vaccinated population in comparison with the SEIR models [10-12] and the SEIQR-SIS model which adds a quarantine population [13]. Other variant consists of the generalisation of such models by incorporating point
and/or distributed delays [8,10-12,14]. All of the aforementioned models are so-called compartmental models since host individuals
are classified depending on their status in relation to the infectious disease. However,
there are diseases where some factors such as the disease transmission, the mortality
rate and so on are the functions of age. Then such diseases are described more precisely
by means of the so-called compartmental models with age structure [15]. Moreover, although the dynamics of infectious diseases transmission through a host
population is continuous-time, some researchers have proposed models composed of difference
equations to describe the dynamics of epidemics and develop treatments to minimise
its effects within the population [16]. On the one hand, a key point in such research works is the choice of an optimal-time
step in order to obtain an acceptable discrete-time model from the discretisation
of the continuous-time ones. On the other hand, an advantage is that discrete-time
models are easier to analyse than continuous-time ones, and then the effectiveness
of a potential treatment to eradicate the disease from the host population can be
easier to derive.

The analysis of the existence of equilibrium points, relative to either the persistence
(endemic equilibrium point) or extinction (disease-free equilibrium point) of the
epidemics in the host population [6,9,11-14], the constraints for guaranteeing the positivity and the boundedness of the solutions
of such models [11,12,17] and the conditions that generate an oscillatory behaviour in such solutions [11,18] have been some of the main objectives in the literature about epidemic mathematical
models. Other important aim is that relative to the design of control strategies in
order to eradicate the persistence of the infection in the host population [2,5,11,12,17]. In this context, an explicit vaccination function of many different kinds may be
added to all aforementioned epidemic models, namely constant [5,12], continuous-time [2,17], impulsive [10], mixed constant/impulsive [11], mixed continuous-time/impulsive [14], discrete-time and so on. Concretely, the research in [17] exhaustively analyses the equilibrium points of an SEIR epidemic model under a vaccination
strategy based on a state feedback control law with respect to the model parameters
and/or the controller gains. The conditions for the eradication of the diseases from
the host population, the extinction of the host population or the persistence of the
disease in a non-extinguished host population are derived form such a study. Other
alternative approaches, as those based on fuzzy rules [19] or networks framework [13,20], have been also proposed for modelling the epidemics transmission through a host
population. In this way, the influence of certain social network parameters such as
visiting probability, hub radius and contact radius on the epidemics propagation has
been investigated [13]. Moreover, there are studies about the influence of the immigration on the persistence
or extinction of the epidemics in a population subject to immigration from other regions
[9]. Also, the influence of epidemic diseases on the dynamics of prey-predator models
has been considered in ecoepidemic models [21].

In this paper, an SEIR epidemic model which includes susceptible (S), infected or
exposed (E), infectious (I) and removed-by-immunity (R) populations is considered.
The dynamics of susceptible and immune populations are directly affected by a vaccination
function , which also has an indirect influence on the time evolution of exposed and infectious
population. In fact, such a vaccination function has to be suitably designed in order
to eradicate the infection from the population. This model has already been studied
in [2] from the viewpoint of equilibrium points in the controlled and free-vaccination cases.
A vaccination auxiliary control law proportional to the susceptible population was
proposed in order to achieve the whole population being asymptotically immune. Such
an approach assumed that the SEIR model was of the aforementioned true-mass action
type, its parameters were known and the illness transmission was not critical. Moreover,
some important issues of positivity, stability and tracking of the SEIR model were
discussed. The main drawback of such a control strategy is the need of online measures
of the susceptible, infected, infectious and immune populations. However, the precise
online measures of susceptible and infected populations are not always feasible in
some real situations, while only true measures of the infectious and whole populations
are available. The main motivation of the present paper is to provide a control solution to overcome
such a drawback. In this sense, the use of a switching control law coupled with a
state observer to synthesise the vaccination function under no precise knowledge of
the exact partial populations which are online estimated by the observer is proposed. Such a law only switches once and in this way the control process is divided into
two stages. In the first stage, the so-called observation stage, the control function
is identically zero and only the observer is working to reduce the initial difference
between the true infectious population measure and the estimated one provided by the
own observer below a prescribe threshold. In the second stage, related to a combined
observation/control stage, the vaccination function is synthesisedby means of an input-output exact feedback linearization technique while the observer
is maintained active providing the estimates of the true partial populations.In both stages, the state observer provides online estimations of susceptible and infectedpopulations through time overcoming the unfeasibility of obtaining true measures of
such partial populations. Such a combination of a linearization control strategy with
a nonlinear observer to online estimate all the partial populations constitutes the
main contribution of the paper. Moreover, mathematical proofs about the epidemics
eradication based on such a controlled SEIR model coupled with the nonlinear observer
are presented while maintaining the non-negativity of all the partial populations
for all time. The exact feedback linearization can be implemented by using a proper nonlinear coordinate
transformation and a static-state feedback control. The use of such a linearization
strategy is motivated by three main facts, namely (i) it is a power tool for controlling
nonlinear systems which is based on well-established technical principles [22,23], (ii) the given SEIR model is highly nonlinear and (iii) such a control strategy
has not been yet applied in epidemic models.

On the one hand, approaches based on switching control laws have been broadly dealt
with in the control theory and its applications [24]. On the other hand, the combination of exact feedback linearization techniques with
state observers has been widely used in many control applications, for instance, in
biological systems and chemical engineering [25,26]. The exact linearization technique requires the system to satisfy some structural
and regularity conditions, like the existence of relative degree, the minimum phase
property and the integrability condition [27,28]. The SEIR epidemic model satisfies such assumptions, and the aforementioned linearization
technique can be applied without any modification. Otherwise, alternative approaches
developed to approximately linearize nonlinear systems violating one or more of such
assumptions could be used [29,30].

The paper is organised as follows. Section 2 describes the set of differential equations
which compound the SEIR model for the propagation of an epidemic disease through a
host population. A result related to the positivity property of such a model is proven.
Section 3 presents a control action based on an input-output linearization technique,
guaranteeing the positivity and stability properties of the system while asymptotically
achieving the eradication of the infection from the host population and, simultaneously,
the whole population becoming immune. The positivity property is required from the
own nature of the system which forbids the existence of negative populations at any
time instant. The control strategy requires the knowledge of the susceptible, infected,
infectious and whole population for all time. In this context, the knowledge of the
infectious and whole population for all time is feasible, but the knowledge of the
susceptible and infected population for all time is not a realistic assumption. As
a consequence, such partial populations have to be estimated by means of an observer
dynamic system. Then a control action based on such estimates, instead of the corresponding
true partial populations, is carried out in Section 4. These theoretical results and
the effectiveness of the feedback input-output linearizing controller combined with
the observer are illustrated by means of some simulation results in Section 5.

Notation is the first open nth real orthant and is the first closed nth real orthant. is a positive real n-vector in the usual sense that all its components are non-negative. Also, and are, respectively, used instead of and for scalars. denotes the identity matrix and the determinant of the matrix M.

2 SEIR epidemic model

Let , , and be, respectively, the susceptible, infected (or exposed), infectious and removed-by-immunity
populations at time t. Consider a time-invariant true-mass action type SEIR epidemic model given by the
following equations:

(2.1)

(2.2)

(2.3)

(2.4)

subject to initial conditions , , and under a vaccination function . In the above SEIR model, is the total population at any time instant , μ is the rate of deaths and births from causes unrelated to the infection, ω is the rate of losing immunity, β is the transmission constant (with the total number of infections per unity of time
at time t being ) and, and are finite and, respectively, the average durations of the latent and infective periods.
All the above parameters are assumed to be non-negative. The total population dynamics
can be obtained by summing-up both sides of (2.1)-(2.4) yielding:

(2.5)

so that the total population is constant . As a consequence, this model is suitable for epidemic diseases with very small mortality
incidence caused by infection and for populations with equal birth and death rates
so that the total population may be considered constant for all time. The following
result relative to the positivity of the SEIR model in the absence of vaccination
is proven. It is relevant since positivity is required for the model validity in real
cases.

Lemma 2.1Assume the SEIR model (2.1)-(2.4) with an initial condition subject toand under no vaccination action before a finite time instant, i.e. . Then.

Proof Let eventually existing finite time instants , , and with being such that:

• If , then and .

• If , then and .

• If , then and .

• If , then and .

Note that either does not exist or it is the first eventual time instant previous to the finite time
instant at which some of the partial populations of the SEIR model reach a zero value and
can be coincident with at most three of its arguments since the total population being
is incompatible with the four partial populations being simultaneously zero. The
remaining of the proof is split into four parts as follows:

(a) Proceed by contradiction by assuming that there exists a finite such that , and (where is the value of the function at the time instant which is infinitesimally close to by the right-hand side) with . Thus, from (2.1) since . The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. The
result contradicts the assumption that and the time instant does not exist.

(b) Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.2). The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such
a result contradicts the assumption that and the time instant does not exist.

(c) Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.3). The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such
a result contradicts the assumption that and the time instant does not exist.

(d) Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.4) since . The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such
a result contradicts the assumption that and the time instant does not exist.

As a result, if and the vaccination function then follows directly since a time instant , for which any of the four partial populations reaches a zero value with its first-time
derivative being simultaneously negative at such a time instant, does not exist. □

Remark 2.1 The result is implied by Lemma 2.1, combined with the equation (2.5), provided that and the SEIR model is initialised such that .

3 Vaccination strategy

An ideal control objective is that the removed-by-immunity population asymptotically
tracks the whole population. In this way, the joint infected plus infectious population asymptotically tends
to zero as time tends to infinity, so the infection is eradicated from the population.
A vaccination control law based on a static-state feedback linearization strategy
is developed for achieving such a control objective. This technique requires a nonlinear
coordinate transformation, based on the theory of Lie derivatives [23], in the system representation.

The dynamics equations (2.1)-(2.3) of the SEIR model can be equivalently written as
the following nonlinear control affine system:

(3.1)

where , and are considered as the state vector, the measurable output signal (i.e. the infectious population) and the input signal of the system , respectively, and is used with

(3.2)

where . The first step to apply a coordinate transformation based on the Lie derivation
is to determine the relative degree of the system. For such a purpose, the following
definitions are taken into account: (i) The kth-order Lie derivative of along is with and (ii) the relative degree r of the system is the number of times that the system output (i.e. the infectious population) must be differentiated in order to obtain the input explicitly,
i.e. the number r such that for and .

From (3.2), , while , so the relative degree of the system is 3 in , i.e. except in the singular surface of the state space where the relative degree is not well defined. Since the relative
degree of the system is exactly equal to the dimension of the state space for any
, the nonlinear coordinate change defined as follows:

(3.3)

allows representing the SEIR model in the so-called normal form in a neighbourhood
of any . Namely

(3.4)

where and

(3.5)

The equations in (3.3) define a mapping whose Jacobian matrix , with for , is non-singular since if . Then the reverse transformation, namely , is available in order to obtain the original state vector from the new one whenever . By direct calculations, such a reverse transformation is given by

(3.6)

Both transformations and are smooth mappings, i.e. they have continuous partial derivatives of any order. Then defines a diffeomorphism on D. The feature that the relative degree of the system is equal to the system order
allows to change it into a linear and controllable one around any point via the coordinate transformation (3.3) and an exact linearization feedback control [23,28]. The following result being relative to the input-output linearization of the system
is established.

Theorem 3.1The state feedback control law defined as

(3.7)

whereforare the controller tuning parameters, induces the linear closed-loop dynamics given by

(3.8)

around any point.

Proof The following state equation for the closed-loop system is obtained:

(3.9)

by introducing the control law (3.7) in (3.4) and taking into account the coordinate
transformation (3.3) and the fact that . Moreover, it follows by direct calculations that

(3.10)

One may express in the state space defined by via the application of the coordinate transformation in (3.6). Then it follows directly
that . Thus, the state equation of the closed-loop system in the state space defined by
can be written as

(3.11)

Furthermore, the output equation of the closed-loop system is with since . From (3.11) and the closed-loop output equation, it follows that

(3.12)

with ℓ denoting the order of the differentiation of . Finally, the dynamics of the closed-loop system (3.8) is obtained by direct calculations
from (3.12). □

Remarks 3.1 (i) The controller parameters , for , will be adjusted so that the roots of the closed-loop system characteristic polynomial
are located at prescribed positions, i.e. for and , with denoting the desired roots of . If one of the control objectives is to guarantee the exponential stability of the
closed-loop system, then all roots of have to be in the open left-half plane, i.e. for all . Then the values , and for the controller parameters have to be chosen in order to achieve such a stability
result. It implies that the strict positivity of the controller parameters is a necessary
condition for the exponential stability of the closed-loop system.

(ii) The control (3.7) may be rewritten as follows:

(3.13)

by using (3.3) and (3.10), or

(3.14)

where is the control parameters vector, by using (3.3) and the facts that and .

(iii) The control law (3.7) is well defined for all except in the surface . However, the infection may be considered eradicated from the population once the
infectious population strictly exceeds zero while it is smaller than one individual.
So the vaccination strategy may be switched off when . This fact implies that the singularity in the control law is not going to be reached,
i.e. such a control law is well defined by the nature of the system. In this sense, the
control law given by

(3.15)

may be used instead of (3.7) in a practical situation. The signal in (3.15) is given by the linearizing control law (3.7) while denotes the eventual time instant after which the infection propagation may be assumed
ended. Formally, such a time instant is defined as follows:

(3.16)

In this way, the control action is maintained active while the infection persists
within the host population and it is switched off once the epidemics is eradicated.

(iv) The linear system (3.8) is strictly identical to the SEIR model (2.1)-(2.4) under
the transformation (3.3) and the control law (3.15) for , i.e. until the time instant at which the epidemics is eradicated.

(v) The implementation of the control law (3.15) requires online measurement of the
susceptible, infected and infectious population. In a practical situation, only online
measures of the infectious and whole populations may be feasible, so the populations
of susceptible and infected can only be estimated. In this context, a complete state
observer is going to be designed for such a purpose in Section 4.

3.1 Controller tuning parameters choice

The application of the control law (3.7), obtained from the exact input-output linearization
strategy, makes the closed-loop dynamics of the infectious population be given by
(3.8). Such a dynamics depends on the control parameters for . Such parameters have to be appropriately chosen in order to guarantee the following
suitable properties: (i) the stability of the controlled SEIR model, (ii) the eradication
of the infection, i.e. the asymptotic convergence of and to zero as time tends to infinity and (iii) the positivity property of the controlled
SEIR model under a vaccination based on such a control strategy. The following theorems
related to the choice of the controller tuning parameter values for are proven, in order to meet such properties under an eventual vaccination effort.

Theorem 3.2Assume that the initial conditionis bounded, and all rootsforof the characteristic polynomialassociated with the closed-loop dynamics (3.8) are of strictly negative real part via an appropriate choice of the free-design controller parametersfor. Then the control law (3.7) guarantees the exponential stability of the transformed controlled SEIR model (3.1)-(3.6) while achieving the eradication of the infection from the host population as time
tends to infinity. Moreover, the SEIR model (2.1)-(2.4) has the following properties: , , andare bounded for all time, , , andexponentially as, and.

Proof The dynamics of the controlled SEIR model (3.8) can be equivalently rewritten with
the state equation (3.11) and the output equation , where , by taking into account that , and . The initial condition in such a realization is bounded since it is related to via the coordinate transformation (3.3), and is assumed to be bounded. The controlled SEIR model is exponentially stable since
the eigenvalues of the matrix A are the roots for of which are assumed to be in the open left-half plane. Then the state vector exponentially converges to zero as time tends to infinity while being bounded for
all time. Moreover, and are also bounded and converge exponentially to zero as from the boundedness and exponential convergence to zero of as according to the first and second equations of the coordinate transformation (3.3).
Then the infection is eradicated from the host population. Furthermore, the boundedness
of follows from that of and , and the fact that the total population is constant for all time. Also, the exponential
convergence of to the total population as is derived from the exponential convergence to zero of and as , and the fact that . Finally, from the third equation of (3.3), it follows that is bounded and it converges exponentially to zero as from the boundedness and convergence to zero of , and as . The facts that and as imply directly that . □

Remark 3.2Theorem 3.2 implies the existence of a finite time instant after which the epidemics is eradicated if the vaccination control law (3.15) is
used instead of that in (3.7). Concretely, such an existence derives from the convergence
of to zero as via the application of the control law (3.7).

Theorem 3.3Assume that an initial condition for the SEIR model satisfies, , i.e. , and, and the constraint. Assume also that some strictly positive real numbersforare chosen such that

(a) , and, so that,

(b) andsatisfy the inequalities:

Then

(i) the application of the control law (3.7) to the SEIR model guarantees that the epidemics is asymptotically eradicated from
the host population while, and, and

(ii) the application of the control law (3.15) guarantees the epidemics eradication after a finite time, the positivity of the controlled SEIR epidemic modeland thatso that,

provided that the controller tuning parametersforare chosen such thatforare the roots of the characteristic polynomialassociated with the closed loop dynamics (3.8).

Proof (i) On the one hand, the epidemics asymptotic eradication is proven by following
the same reasoning as in Theorem 3.2. On the other hand, the dynamics of the controlled SEIR model (3.8) can be written
in the state space defined by as in (3.11). From such a realization, taking into account the first equation in
(3.3) and the fact that for are the eigenvalues of A, it follows that

(3.17)

for some constants for being dependent on the initial conditions , and . In turn, such initial conditions are related to the initial conditions of the SEIR
model in its original realization, i.e. in the state space defined by via (3.3). The constants for can be obtained by solving the following set of linear equations:

(3.18)

where (3.3) and (3.17) have been used. Such equations can be more compactly written
as , where

(3.19)

Once the desired roots of the characteristic equation of the closed-loop dynamics
have been prefixed, the constants for of the time-evolution of are obtained from since is a non-singular matrix, i.e. an invertible matrix. In this sense, note that since is the Vandermonde matrix [31] and the roots for have been chosen different among them. Namely

(3.20)

where the functions and are defined as follows:

(3.21)

In particular, since , , , , , and by taking into account the constraints in (a). On the one hand, is proven directly from (3.17) as follows. One ‘a priori’ knows that . However, the sign of both and may not be ‘a priori’ determined from the initial conditions and constraints in (a). The following four
cases may be possible: (i) and , (ii) and , (iii) and , and (iv) and . For the cases (i) and (ii), i.e. if , it follows from (3.17) that

(3.22)

where the facts that and, and since , have been taken into account. For the case (iii), i.e. if and , it follows from (3.17) that

(3.23)

by taking into account that , since and the fact that

(3.24)

where (3.20), (3.21), , and the constraints in (a) and (b) have been used. In particular, the coefficient
multiplying to in (3.24) is non-negative if and satisfy the third inequality of the constraints (b) by taking into account and . This later inequality is directly implied by , , , and . Finally, for the case (iv), i.e. if and , it follows from (3.17) that

(3.25)

where the constraints , and , since , have been taken into account. In summary, if all partial populations are initially non-negative and the roots for of the closed-loop characteristic polynomial satisfy the constraints in (a) and (b).
On the other hand, one obtains by direct calculations from (3.6) and (3.17) that

(3.26)

by taking into account that and . If one fixes the parameter then

(3.27)

where the fact that the function defined by

(3.28)

is zero for has been used. From the first equation in (3.27), it follows that and then

(3.29)

by applying such a relation between and in (3.27) and by taking into account that , and since . In this way, the non-negativity of has been proven. From the second equation in (3.27), it follows that and then

(3.30)

by applying such a relation between and in (3.27) and by taking into account that since , , , and and since . In this way, the non-negativity of has been proven. Note that the function defined by (3.28) is an upper-open parabola zero-valued for and so from the assumption that .

(ii) On the one hand, if the control law (3.15) is used instead of that in (3.7),
then the time evolution of the infectious population is also given by (3.17) while
the control action is active. Thus, the exponential convergence of to zero as in (3.17) implies directly the existence of a finite time instant at which the control (3.15) switches off. Obviously, the non-negativity of , and is proven by following the same reasoning used in the part (i) of the current theorem.
The non-negativity of is proven by using continuity arguments. In this sense, if reaches negative values for some starting from an initial condition , then passes through zero, i.e. there exists at least a time instant such that . Then it follows from (2.4) that

(3.31)

by introducing the control law (3.15) and taking into account the facts that and since has been used. Moreover, the non-negativity of , and as it has been previously proven, implies that , and . Also, since and from the definition of in (3.16). Then one obtains

(3.32)

from (3.31). The controller tuning parameters for are related to the roots for of the closed-loop characteristic polynomial , see Remark 3.1 (i), by

(3.33)

The assignment of for such that the constraints in (a) and (b) are fulfilled implies that

(3.34)

Then by taking into account (3.34) in (3.32). The facts that , and imply that via complete induction. Finally, the positivity of the controlled SEIR model follows from the non-negativity of , , and and Lemma 2.1.

On the other hand, it follows from (3.13) and (3.15) that

(3.35)

by taking into account that . Moreover,

(3.36)

where the facts that , , and have been used. If the roots of the polynomial satisfy the conditions in (a) and (b), it follows from (3.36) that

(3.37)

by taking into account the third equation in (3.34) and the non-negativity of , and . Finally, it follows that from (3.15) and (3.37). □

In summary, this section has dealt with a vaccination strategy based on linearization
control techniques for nonlinear systems. The proposed control law satisfies the main
objectives required in the field of epidemics models, namely the stability, the positivity
and the eradication of the infection from the population. Such results are proven
formally in Theorems 3.2 and 3.3. In Section 5, some simulation results illustrate the effectiveness of such a vaccination
strategy. However, such a strategy has a main drawback, namely the control law needs
the knowledge of the true values of the susceptible, infected and infectious populations
at all time instants which are not available in certain real situations. An alternative
approach useful to overcome such a drawback is dealt with in the following section
where an observer to estimate all the partial populations is proposed.

4 Vaccination control strategy based on the use of a state observer

The control laws (3.7), or equivalently (3.13) or (3.14), and (3.15) require the online
measurement of all the state variables, namely , and . However, the online measures of the infected and susceptible populations are rarely
affordable in certain real situations where only knowledge about the infectious and
total populations may be available. As a consequence, the control laws (3.7) and (3.15)
may not be implemented. An alternative approach involving the use of a complete state
observer is proposed. This observer provides online estimates, and of the true state variables. Such estimates are used instead of , and for the implementation of the switching control law given by

(4.1)

with defined as follows:

(4.2)

where denotes the estimate of the state vector corresponding to the system representation (3.4)-(3.5). The switching time instants
in the control law (4.1), i.e. and are defined as follows:

(4.3)

for some real constants , and so that . Note that and . The signal denotes the estimation error corresponding to the infectious population, i.e. the deviation between the infectious population estimated by the observer and the
true one. Note that from , and then also , by taking into account the coordinate change (3.3) or (3.6). In other words, the
estimation error associated to the infectious population is identical in both system
representations, defined, respectively, by (3.1)-(3.2) and (3.4)-(3.5). The signal
(4.2) has the same structure as the control law (3.14) used for linearizing the SEIR
model in the case where the measures of all partial populations were available. Note
that the control law (4.1)-(4.3) is expressed in terms of the variables of since the observer design is developed based on the system representation (3.4)-(3.5).
Moreover, the observer has to be designed in such a way that the estimation error
converged rapidly to zero while maintaining the stability, positivity and epidemics
eradication objectives.

The SEIR model (3.1)-(3.2) is diffeomorphic on D to the system (3.4)-(3.5) by applying the nonlinear coordinate transformation (3.3). In the system representation (3.4)-(3.5) the functions and fit into the called normal form given by

(4.4)

with , , , defined as (3.5), and . The existence of such a diffeomorphism implies that the SEIR model is uniformly observable on D for any input in view of Theorem 2 of [25]. This property allows constructing an observer in the coordinates corresponding to
the state representation (3.4)-(3.5). The state equation of such an observer is as
follows:

(4.5)

with an initial condition . The matrix is the unique positive-definite symmetric solution of the algebraic Lyapunov equation
below

(4.6)

where is a tuning parameter, referred to as the observer gain, and Δ and C are the following matrices:

(4.7)

The following result relative to the existence of a finite time instant defined as in (4.3) is proven.

Lemma 4.1Assume that

(i) The SEIR model parameters are such that,

(ii) in the definition of the switching time instantsatisfies,

(iii) the control parameterin (4.2) and the constantin the definition ofare such that, and

(iv) the observer gainis large enough for the estimation error, withand, to converge asymptotically to zero as.

Then a finite time instantat which the control law (4.1) switches for the first time exists.

Proof On the one hand, note that the functions and in (4.4) are globally Lipschitz [32] on . Then the observer (4.5) for the SEIR model is well defined in the sense that it
guarantees the asymptotic convergence to zero of the estimation error as provided that is uniformly bounded and the gain is large enough in view of Theorem 3 of [25]. Such a result implies that

(4.8)

for some definite positive function if is large enough since . Then there exists a finite time instant such that for , with being the eventual time instant at which the control law switches for the first time
and any real constant. Note that and . Furthermore, the convergence rate of depends on the value of the observer gain θ in the sense that such a convergence rate is increased as the observer gain increases.

The existence of the finite time instant at which the control law switches is demonstrated below ad absurdum. In this sense,
suppose that there are no time instants such that and for some real constants and satisfying the constraints (ii) and (iii) respectively. Then from (4.1). As a consequence, on the one hand, the SEIR model converges to its endemic
equilibrium point defined by the following partial populations [2]:

(4.9)

since the fact that the SEIR parameters fulfil the condition has been assumed. The convergence of to as time tends to infinity implies directly the existence of a finite time instant
such that since in view of constraint (ii). On the other hand, if , then tends asymptotically to zero as time tends to infinity from (4.8). As a consequence,
converges asymptotically to as time tends to infinity. Furthermore, converges to , which denotes the endemic equilibrium point in the state space realization defined
by , as time tends to infinity from the convergence of the SEIR model to its endemic
equilibrium point. Then also converges to as time tends to infinity. As a consequence, from (4.2) , converges to a value given by

(4.10)

provided that the parameters and satisfy the constraint (iii) and by taking into account that , , and from (3.3), (3.5) and (4.9). Thus, (4.10) shows the existence of a finite time instant
such that . As a consequence, there exists a finite time instant at which the control law switches, which contradicts the starting hypothesis. □

Remark 4.1Lemma 4.1 requires the SEIR model parameters to fulfil the condition so that the endemic equilibrium point exists. Otherwise, i.e. if the parameters are such that , the SEIR model converges to the disease-free equilibrium point at which all the population is susceptible and no individual is either infected or
infectious so a control action is not necessary to eradicate the epidemic. Then such
a situation is not interesting from the control theory viewpoint.

The following result, supported by Lemma 2.1, Theorem 3.3 and Lemma 4.1, is relative to the eradication of the epidemic via the application of the control law (4.1)-(4.3) while guaranteeing the non-negativity
of the control signal for all time.

Theorem 4.1Assume that

(i) The SEIR model parameters are such that,

(ii) its initial condition satisfies,

(iii) in the definition of the switching time instantand the SEIR model parameters are such that, withlarge enough,

(iv) in the definition of the switching time instantis small enough forand, and,

(v) the gain observeris large enough, and

(vi) the controller parametersfor, which are related via (3.33) to the rootsforof the closed-loop dynamics characteristic polynomial, are chosen such that the conditions (a)-(b) of Theorem 3.3 are fulfilled whilefor some sufficient large real constant.

Then the control law (4.1)-(4.3) combined with the observer (4.4)-(4.7) leads to the eradication of the epidemics while guaranteeing the non-negativity of the control signal.

Proof By substituting the solution of (4.6) in (4.5), one can write the observer state
equation as follows:

(4.11)

since and . The state equation of the SEIR model in the state space realization (3.4)-(3.5)
can be rewritten as follows:

(4.12)

Then the error dynamics between (4.11) and (4.12) becomes

(4.13)

Let be a Lyapunov function candidate. Note that (with if and only if ) since L is a positive definite symmetric matrix. The time derivative of such a function is

(4.14)

where

(4.15)

After the control law switches at the time instant , the closed-loop dynamics of the observer is given by

(4.16)

where the control law (4.1)-(4.3) has been introduced in (4.11) and denotes the eventual time instant at which the epidemics is eradicated.

On the one hand, the definition of and continuity arguments of the functions , and imply that there exits some definite-positive, monotone function , decreasing with and satisfying , such that , and since from the assumption (iv) of the current theorem. Furthermore, Lemma 2.1 and the definition of in (4.3) guarantee that

(4.17)

where the coordinates change (3.3) and the fact that has been used. By using continuity arguments of the functions , , and , there exists some definite-positive, monotone function increasing with , decreasing with and satisfying , such that , , and with . Then one obtains that

(4.18)

, with , by taking into account the definition of in (3.5) and where

(4.19)

with

(4.20)

Finally, the definition of also implies that . Then by using continuity arguments of the function , or continuity arguments of , there exists some definite-positive, monotone function , increasing with and satisfying , such that . Then from (4.14), (4.15) and (4.18)-(4.20), it follows that

(4.21)

with , if the observer gain θ is large enough for both and to be guaranteed . In this context, on the one hand, note that all terms of in (4.15), except the one with , are strictly negative whenever , and for any observer gain . Such a term may be made small enough, in comparison with the absolute value of the contribution
of the rest of the terms in , by means of a large enough observer gain so that . On the other hand, can be also ensured with a large enough observer gain . In this case, the term depending on in (4.21) is strictly negative since in such a time interval. Note also that , which is non-negative , may be made small in comparison with the absolute value of the rest of the terms
in (4.21) by using a sufficiently large so that .

From (4.21) and the definition of the Lyapunov function candidate , if follows that is monotone decreasing and that is also monotone decreasing with from the definition of the time instant . Then from (4.16), it follows that

(4.22)

since is the absolute value of the dominant eigenvalue of the state matrix A in (4.16), according to condition (vi) of the theorem, and where is some upper-bound of the transition matrix associated to the system (4.16). Moreover,
for some real constants and from the fact that is monotone decreasing . The constant is related to the observer gain in the sense that ρ can be as large as θ. Then from (4.22), one obtains that

(4.23)

provided that . Note that if satisfies , then any also satisfies it. From (4.23), the population estimates vector would converge asymptotically to zero as . As a consequence, if the time interval is sufficiently large for the conditions (4.17), (4.18) and , (4.21) be satisfied until reaching a time instant at which , then the propagation of the infection will be eradicated from the population by
taking into account that is very small, while maintaining the positivity of the controlled SEIR model. □

Remarks 4.1 (i) The observer dynamics consists of a copy of that of the system model in the transformed
state representation with an additional term proportional to the observation error . Moreover, such a corrective term does not depend on the system realization, but
only on the dimension of the state space since according to the coordinate transformation (3.3).

(ii) The control strategy described by (4.1)-(4.3) is composed of two consecutive
stages. The first one is an observation stage at which no control action is applied
to the SEIR model and the main objective is that the observer variables go converging
to the true partial populations. Once the estimation error is sufficiently small,
such that the observer variables track the true population with a suitable precision,
this observation stage ends. Then the second stage at which a control action is applied
begins. Such a control action linearizes the closed-loop observer so that the estimated
variables converge asymptotically to zero as time grows while also guaranteeing the
convergence to zero of the estimation error. This implies that the true infectious
and infected populations decrease toward zero as time grows until reaching a finite
time instant when the epidemic disease is considered eradicated from the host population. After
such an instant, the control action is removed from the system.

(iii) On the one hand, the length of the observation stage may be short enough with
a suitable choice of the observer gain. In this sense, such a length decreases as
the value of the observer gain θ increases. Then a large value for θ seems to be appropriate. On the other hand, a large value of θ makes the contribution of the perturbation term in (4.16) considerable. As a consequence, a tradeoff value for θ has to be searched. In this context, in the limiting case that θ was a very small positive real number (i.e.) the closed-loop dynamics of the observer system would be with A defined in (3.11), i.e. such a dynamics would be identical to that of the closed loop SEIR model without
the observer. As a consequence, if the controller parameters for were chosen according to the conditions of Theorem 3.3, the estimates in would be non-negative and would tend asymptotically to zero as time tends to infinity. However, the estimation
error and would not converge asymptotically to zero as time tends to infinity with such a value
of , so the infection would not be eradicated. On the contrary, if the value for were very large, then a very small real constant would be necessary for the term , which acts as a perturbation for the observer closed-loop system (4.16), to be sufficiently
small. Such a fact would delay the beginning of the control action.

(iv) Once the observer has been designed in the state representation , associated to the state vector , the coordinate transformation (3.6) has to be applied to obtain the estimates in
, corresponding to the original state vector , from the vector . The following dynamics equations for the estimated populations are obtained from
(4.11) by applying (3.6)

(4.24)

Such equations can be written more compactly as follows:

(4.25)

where . The definitions of and in (3.2) have been taken into account, and also that

(4.26)

Note that the observer state equation is a copy of the SEIR model (3.1)-(3.2) with
an additional term depending on the output observation error and on the observer gain.

(v) The equation (4.21) is a key result in the mathematical proof of stability. Note
that once the control law switches at time instant , and the system passes from the observation stage to the observation/control one,
it is ensured that the observation error still exhibits an asymptotically decaying
behaviour on a sufficiently large time interval provided that the size of the control signal is large enough at such a switching
time instant. This may be guaranteed by means of a sufficient large real constant
, and simultaneously and with small enough and large enough so that and on in order to ensure the validity of (4.18) on .

(vi) In view of (3.33), the assumption (vi) of Theorem 4.1 can be satisfied by choosing a root for the closed-loop system characteristic polynomial with as large as it is necessary for the constraints (a) and (b) of Theorem 3.3 to be fulfilled.

5 Simulation results

An example based on an outbreak of influenza in a British boarding school in early
1978 [4] is used to illustrate the theoretical results presented in the paper. Such an epidemic
can be described by the SEIR mathematical model (2.1)-(2.4) with the parameter values:
, , and . A total population of is considered with the initial conditions , , and . Three sets of simulation results are presented to compare the evolution of the SEIR
mathematical model populations in three different situations, namely (i) when no vaccination
control actions are applied (ii) if a control action based on the feedback input-output
linearization approach is applied and (iii) if a nonlinear observer dynamics is coupled
with a switching control law, as that defined by (4.1)-(4.3), so as to use population
estimates in the vaccination control.

5.1 Epidemic evolution without vaccination

The time evolution of the respective populations is displayed in Figure 1. The model tends to its endemic equilibrium point as time tends to infinity. There
are susceptible, infected and infectious populations at such an equilibrium point.
As a consequence, the control action has to be applied in order to eradicate the epidemics.

Figure 1.Time evolution of the individual populations without vaccination.

5.2 Epidemic evolution with the feedback control law without an observer

The control law given by (3.15)-(3.16) is applied with and the free-design controller parameters , for , so that the roots of the characteristic polynomial associated with the closed-loop dynamics (3.8) are , and . Such values for are obtained from (3.33). The time evolution of the respective populations is displayed
in Figure 2 and the vaccination function in Figure 3. The vaccination control action achieves the control objectives as it is seen in
Figure 2. In this sense, the infection is eradicated from the population since both infectious
and infected populations converge rapidly to zero. Also, the susceptible population
converges to zero while the removed-by-immunity population tracks asymptotically the
whole population as time tends to infinity. Such a result is coherent with the result
proven in Theorem 3.3. Moreover, the positivity of the system is maintained for all time as it can be seen
from the figures. Such a property is satisfied although all constraints of the assumption
(b) of Theorem 3.3 are not fulfilled by the system parameters and the chosen control parameters. However,
such a result is coherent since such constraints are sufficient but not necessary
to prove the positivity of the system. The switched off time instant for the vaccination
control action is .

Figure 2.Time evolution of the individual populations with a vaccination control action.

5.3 Epidemic evolution with the feedback control law with an observer

The observer initial condition is , , . The observer gain is and the switching control law (4.1)-(4.3), with the tuning controller parameters
being fixed as in the previous Section 5.2, is applied. The parameters and are considered in the switching control law. The time evolutions of the true infectious,
infected and susceptible populations with their corresponding estimates given by the
observer are displayed in Figures 4, 5 and 6, respectively. The fast convergence of the estimates to their corresponding true
values can be seen. This fact allows the use of the estimated variables instead of
the true ones to implement the vaccination control law which leads to the eradication
of the infection. In this sense, the asymptotic convergence of the infectious, infected
and susceptible populations to zero as time tends to infinity can be seen from the
figures. Figures 7 and 8 display the time evolutions of the removed-by-immunity population and the vaccination
function issued by the controller coupled with the observer system, respectively.
On the one hand, the removed-by-immunity population converges asymptotically to the total population
as time tends to infinity. On the other hand, the vaccination function is positive for all time. As these are
all partial populations, the positivity of the controlled SEIR-model combined with
the observer is preserved. In the first days, approximately until the sixth day as
it can be seen in Figure 8, none vaccination action is applied. This time period corresponds to the observation
stage when the observer is working to reduce the initial estimation error. In this
way, appropriate estimates of the true infectious, infected and susceptible to the
infection populations are obtained. The estimates are used in the control law applied
in the observation/control stage in order to eradicate the infection from the host
population. Note that the observation stage is short compared to the observation/control
stage; thus the vaccination action can begin a few days later than the infection is
detected in the host population. Also, the vaccination function takes a large value at the switching time instant of the control law (4.1)-(4.3), approximately as it can be seen in Figure 8. Then the time interval , where the vaccination control is ensured positive and then the estimation errors
maintain the behaviour of asymptotic convergence to zero as time grows, is large enough
to eradicate the infection. Moreover, such a convergence behaviour guarantees the
positivity of the vaccination function as it was proven in Theorem 4.1. The switched off time instant for the vaccination control action is as it is observed from Figures 4, 5 and 6.

Figure 4.Time evolution of the infectious population and its estimate.

5.3.1 Influence of the observer gain in the time evolution of the system

Three different values for the observer gain are considered, namely , and . The time evolutions of the estimation errors associated to the infectious, infected
and susceptible to the infection populations for the three values of θ are displayed in Figures 9, 10 and 11, respectively. One can see that the convergence rate to zero of the three estimation
errors increases as the value for θ is increased. Figure 12 displays the time evolution of the immune population for the three different values
of the observer gain. This figure illustrates that the infection is eradicated from
the host population for any of the three values of the observer gain since the whole
population asymptotically becomes immune as time tends to infinity. Moreover, the
whole population becomes immune faster as the observer gain increases. In other words,
the epidemic is eradicated earlier as the observer gain increases. Finally, Figure
13 displays the time evolution of the vaccination function for different values of θ. One can deduce from such a figure that the switching time in the vaccination function happens earlier as the observer gain increases. Also,
the figure shows that the vaccination function converges asymptotically to the same
value as time grows (approximately ), for any of the three considered observer gains, until the control action is removed
at . The tradeoff between observer precision and speed in the epidemics eradication to
choose the observer gain, as it was commented in Remarks 4.1(iii), can be deduced from Figures 9 to 13. On the one hand, as the observer gain increases, the peak in the estimation errors
increases, which can be clearly seen in Figures 10 and 11. Then the precision in the observer variables becomes worse as the observer gain
increases. On the other hand, the epidemic eradication is achieved earlier as the
observer gain increases as it can be seen in Figure 12. This is due to the fact that the estimation errors converge faster to zero as the
observer gain increases (see Figures 10, 11 and 12), and then the switching time happens earlier and the vaccination action consequences are reached more rapidly.

6 Concluding remarks

A vaccination control strategy based on feedback input-output linearization techniques
has been proposed to fight against the propagation of epidemic diseases within a host
population. An SEIR model with known parameters is used to describe the propagation
of the disease. The stability and positivity properties of the closed-loop system
have been proven in the case where true data of the susceptible, infected and infectious
populations are available. Otherwise, the control law may be combined with a complete
state non-linear observer which provides online estimates of such populations used
in the vaccination controls. These theoretical results are complemented with some
simulation results to illustrate the effectiveness of the proposed approach. Future
research into the subject is going to deal with the application of the current approach
and similar non-linear techniques to other disease propagation models.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this paper. All authors
read and approved the final manuscript.

Acknowledgements

The authors thank the Spanish Ministry of Education for its support of this work through
grants DPI2009-07197 and DPI2012-30651 and to the Basque Government for its support
through grants IT378-10, SAIOTEK SPE07UN04 and SAIOTEK SPE09UN12.